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Properties

Label 2-644-161.100-c1-0-9
Degree $2$
Conductor $644$
Sign $0.741 - 0.670i$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.81 + 2.54i)3-s + (−0.461 + 0.0890i)5-s + (2.62 − 0.356i)7-s + (−2.21 − 6.39i)9-s + (4.09 − 3.22i)11-s + (4.64 − 2.98i)13-s + (0.610 − 1.33i)15-s + (0.933 + 0.890i)17-s + (0.893 − 0.851i)19-s + (−3.84 + 7.31i)21-s + (4.20 + 2.29i)23-s + (−4.43 + 1.77i)25-s + (11.2 + 3.31i)27-s + (−8.07 + 2.37i)29-s + (2.17 − 0.207i)31-s + ⋯
L(s)  = 1  + (−1.04 + 1.46i)3-s + (−0.206 + 0.0398i)5-s + (0.990 − 0.134i)7-s + (−0.737 − 2.13i)9-s + (1.23 − 0.971i)11-s + (1.28 − 0.828i)13-s + (0.157 − 0.345i)15-s + (0.226 + 0.215i)17-s + (0.204 − 0.195i)19-s + (−0.838 + 1.59i)21-s + (0.877 + 0.479i)23-s + (−0.887 + 0.355i)25-s + (2.17 + 0.637i)27-s + (−1.50 + 0.440i)29-s + (0.389 − 0.0372i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.741 - 0.670i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ 0.741 - 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15064 + 0.443034i\)
\(L(\frac12)\) \(\approx\) \(1.15064 + 0.443034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.62 + 0.356i)T \)
23 \( 1 + (-4.20 - 2.29i)T \)
good3 \( 1 + (1.81 - 2.54i)T + (-0.981 - 2.83i)T^{2} \)
5 \( 1 + (0.461 - 0.0890i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-4.09 + 3.22i)T + (2.59 - 10.6i)T^{2} \)
13 \( 1 + (-4.64 + 2.98i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-0.933 - 0.890i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-0.893 + 0.851i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (8.07 - 2.37i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-2.17 + 0.207i)T + (30.4 - 5.86i)T^{2} \)
37 \( 1 + (1.94 + 5.63i)T + (-29.0 + 22.8i)T^{2} \)
41 \( 1 + (-3.48 - 4.02i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (4.67 + 10.2i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + (0.0169 - 0.0293i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.310 + 6.50i)T + (-52.7 - 5.03i)T^{2} \)
59 \( 1 + (-10.5 - 5.42i)T + (34.2 + 48.0i)T^{2} \)
61 \( 1 + (-4.48 - 6.29i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (-13.9 + 5.58i)T + (48.4 - 46.2i)T^{2} \)
71 \( 1 + (1.91 - 13.3i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.83 - 11.6i)T + (-64.8 + 33.4i)T^{2} \)
79 \( 1 + (0.186 + 3.91i)T + (-78.6 + 7.50i)T^{2} \)
83 \( 1 + (1.47 - 1.70i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (11.8 + 1.12i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-0.235 - 0.272i)T + (-13.8 + 96.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95122767242418982038612940346, −9.959206140727116328989848754464, −9.007029987426281235351841090950, −8.355736192848022251026285539454, −6.93641329897460804033552253758, −5.64630310442700087915633263146, −5.43468952331260042716195443768, −3.91899382042508983244820673433, −3.65482822105781276338208670040, −1.02273790335720402957773505409, 1.21744387158575426776220445502, 1.96070917322022380551230118222, 4.07543158680360907593252080570, 5.12922170407327324977368450000, 6.19339376524839754756373039873, 6.80936564787452245333557722051, 7.66550311702197197334496563347, 8.454359689676612824734282185334, 9.552224691986630009381216289245, 10.98888986673014523868159260193

Graph of the $Z$-function along the critical line